Research Article Toeplitz Operators with Lagrangian Invariant Symbols Acting on the Poly-Fock Space of ℂ n Jorge Luis Arroyo Neri , Armando Sánchez-Nungaray , Mauricio Hernández Marroquin , and Raquiel R. López-Martínez Faculty of Mathematics of Universidad Veracruzana, Mexico Correspondence should be addressed to Armando Sánchez-Nungaray; armsanchez@uv.mx Received 25 March 2021; Accepted 8 October 2021; Published 25 November 2021 Academic Editor: Humberto Rafeiro Copyright © 2021 Jorge Luis Arroyo Neri et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We introduce the so-called extended Lagrangian symbols, and we prove that the C ∗ -algebra generated by Toeplitz operators with these kind of symbols acting on the homogeneously poly-Fock space of the complex space ℂ n is isomorphic and isometric to the C ∗ -algebra of matrix-valued functions on a certain compactification of ℝ n obtained by adding a sphere at the infinity; moreover, the matrix values at the infinity points are equal to some scalar multiples of the identity matrix. 1. Introduction Let m ∈ ℕ, the one-dimensional m poly-Fock space F 2 m ðℂÞ ⊂ L 2 ðℂ, dμÞ consists of all m-analytic functions φ which satisfy ∂ ∂ z  m φ = 1 2 m ∂ ∂x + i ∂ ∂y   m φ = 0, ð1Þ where dμ = π −1 e −z·  z dxdy is the Gaussian measure in ℂ and d xdy is the Euclidian measure in ℝ 2 = ℂ. Further, the one- dimensional true poly-Fock space of order m is given by F 2 m ðÞ ℂ ðÞ = F 2 m ℂ ðÞ⊝F 2 m−1 ℂ ðÞ: ð2Þ In the case of several variables, for n ∈ ℕ, the n-dimensional Gaussian measure in ℂ n is given by dμ n ðzÞ = π −n e −jzj 2 dxdy, where dxdy is the Euclidian measure in ℝ 2n . We have that the space L 2 ðℂ n , dμ n Þ is the tensorial product of n components L 2 ℂ n , dμ n ð Þ = L 2 ℂ, dμ ðÞ ⊗⋯⊗ L 2 ℂ, dμ ðÞ, ð3Þ and the Fock space F 2 ðℂ n Þ is F 2 ℂ n ðÞ = F 2 ℂ ðÞ ⊗⋯⊗ F 2 ℂ ðÞ: ð4Þ Given a multi-index α = ðα 1 , ⋯, α n Þ ∈ ℤ n + , the poly-Fock space F 2 α ðℂ n Þ of order α is given by F 2 α ℂ n ðÞ = ⊗ j=1 n F 2 α j ℂ ðÞ: ð5Þ Similarly, the true poly-Fock space F 2 ðαÞ ðℂ n Þ is F 2 α ðÞ ℂ n ðÞ = ⊗ j=1 n F 2 α j ðÞ ℂ ðÞ: ð6Þ In [1], Vasilevski introduced the poly-Fock spaces over ℂ n and he obtained the following decomposition formula: L 2 ℂ n , dμ n ð Þ = ⊕ α jj=n ∞ F 2 α ðÞ ℂ n ðÞ: ð7Þ Moreover, he showed that the true poly-Fock space F 2 ðαÞ ðℂ n Þ is isomorphic and isometric to L 2 ðℝ n , dxÞ ⊗ ~ H α−1 , where ~ H α−1 is the one-dimensional space generated by the function Hindawi Journal of Function Spaces Volume 2021, Article ID 9919243, 13 pages https://doi.org/10.1155/2021/9919243